The arithmetic sequence $a_i$ is defined by the formula: $a_1 = 2$ $a_i = a_{i - 1} -3$ Find the sum of the first $335$ terms in the sequence.
Answer: Getting started Let's write out the first few terms of the series: $2 + (-1) + (-4) + (-7)...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $3$ less than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {2})$ and the number of terms $(n = {335})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $335 -1= 334$ terms after the first term. The sequence decreases by $3$ for each new term. So, the sequence decreases by a total of $334 \cdot 3 = 1002$ from where it starts at $2$. That means the last term must be $2-1002 = {-1000}$. In other words, $a_n = {-1000}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{335}}&= \dfrac {\left({2} + ({-1000}) \right)}{2} \cdot {335} \\\\ S_{{335}} &= -499 \left(335\right) \\\\ S_{{335}} &= -167{,}165\end{aligned}$ The answer $ -167{,}165 $